## 15379 Reputation

12 years, 110 days

## Of cause, it's way more difficult, but i...

Of cause, it's way more difficult, but i don't know different way.

## Of cause, it's way more difficult, but i...

Of cause, it's way more difficult, but i don't know different way.

## @Markiyan Hirnyk  You have given a ...

@Markiyan Hirnyk  You have given a definition by reference to Wikipedia: "A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. " That is the definition I had in mind. If the condition is imposed only on the increase in the first row, the number of Latin squares is 8:

n:=0: NList:=[]:

for i from 1 to 960 do

if [seq(List[i][1,k], k=1..5)]=[1,2,3,4,5] then n:=n+1: NList:=[op(NList), List[i]]: fi:

od:

n;

op(NList[1..4]);

op(NList[5..8]);

## @Markiyan Hirnyk  You have given a ...

@Markiyan Hirnyk  You have given a definition by reference to Wikipedia: "A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. " That is the definition I had in mind. If the condition is imposed only on the increase in the first row, the number of Latin squares is 8:

n:=0: NList:=[]:

for i from 1 to 960 do

if [seq(List[i][1,k], k=1..5)]=[1,2,3,4,5] then n:=n+1: NList:=[op(NList), List[i]]: fi:

od:

n;

op(NList[1..4]);

op(NList[5..8]);

## This is obvious...

@Markiyan Hirnyk  The number of normalized Latin squares with this property is 0, which is obvious without any calculations.

## This is obvious...

@Markiyan Hirnyk  The number of normalized Latin squares with this property is 0, which is obvious without any calculations.

## Another exceptional case...

restart:

Eq:=((diff(R(r), r, r, r, r))*r^4+3*(diff(R(r), r))*r-3*(diff(R(r), r, r))*r^2+2*(diff(R(r), r, r, r))*r^3-3*R(r)+4*R(r)*n+2*R(r)*n^2-4*(diff(R(r), r))*r*n+2*(diff(R(r), r))*r*n^2+4*(diff(R(r), r, r))*r^2*n-2*(diff(R(r), r, r))*r^2*n^2-4*R(r)*n^3+R(r)*n^4)/r^4 = (-108-1362*n+2122*n^2+2019*n^3-3032*n^4+401*n^7+1192*n^6-1033*n^5-25*n^9+6*n^10-180*n^8-4128*r^(n+3)*n^3+2304*r^(n+3)*n-576*r^(n+5)*n-760*n^6*r+575*n^5*r-244*n^7*r-2108*n^3*r-11616*r^(n+3)*n^4-5280*r^(n+3)*n^5-456*n^2*r-6912*r^(3*n+3)*n^3+192*r^(n+3)*n^7-6*n^10*r+152*n^8*r+13*n^9*r-192*r^(n+5)*n^7+4224*r^(3*n+3)*n^2-9216*r^(3*n+3)*n^5-16512*r^(3*n+3)*n^4-1632*r^(n+5)*n^6-1536*r^(3*n+3)*n^6-7680*r^(n+5)*n^3-8832*r^(n+5)*n^4-3360*r^(n+5)*n^2-96*r^(n+3)*n^6+2304*r^(3*n+3)*n-5376*r^(n+5)*n^5+2546*n^4*r+4800*r^(n+3)*n^2)/(576*r^4+192*r^4*n^4+1248*r^4*n^3+2688*r^4*n^2+2208*r^4*n):

Sol:=value(dsolve(Eq)):

expand(simplify(subs(n=0, Sol)));

n:=0: expand(dsolve(Eq));

## Another exceptional case...

restart:

Eq:=((diff(R(r), r, r, r, r))*r^4+3*(diff(R(r), r))*r-3*(diff(R(r), r, r))*r^2+2*(diff(R(r), r, r, r))*r^3-3*R(r)+4*R(r)*n+2*R(r)*n^2-4*(diff(R(r), r))*r*n+2*(diff(R(r), r))*r*n^2+4*(diff(R(r), r, r))*r^2*n-2*(diff(R(r), r, r))*r^2*n^2-4*R(r)*n^3+R(r)*n^4)/r^4 = (-108-1362*n+2122*n^2+2019*n^3-3032*n^4+401*n^7+1192*n^6-1033*n^5-25*n^9+6*n^10-180*n^8-4128*r^(n+3)*n^3+2304*r^(n+3)*n-576*r^(n+5)*n-760*n^6*r+575*n^5*r-244*n^7*r-2108*n^3*r-11616*r^(n+3)*n^4-5280*r^(n+3)*n^5-456*n^2*r-6912*r^(3*n+3)*n^3+192*r^(n+3)*n^7-6*n^10*r+152*n^8*r+13*n^9*r-192*r^(n+5)*n^7+4224*r^(3*n+3)*n^2-9216*r^(3*n+3)*n^5-16512*r^(3*n+3)*n^4-1632*r^(n+5)*n^6-1536*r^(3*n+3)*n^6-7680*r^(n+5)*n^3-8832*r^(n+5)*n^4-3360*r^(n+5)*n^2-96*r^(n+3)*n^6+2304*r^(3*n+3)*n-5376*r^(n+5)*n^5+2546*n^4*r+4800*r^(n+3)*n^2)/(576*r^4+192*r^4*n^4+1248*r^4*n^3+2688*r^4*n^2+2208*r^4*n):

Sol:=value(dsolve(Eq)):

expand(simplify(subs(n=0, Sol)));

n:=0: expand(dsolve(Eq));

## @Markiyan Hirnyk Do you know how ma...

@Markiyan Hirnyk Do you know how many more of these exceptional cases?

Here's another example:

restart:

Eq:=((diff(R(r), r, r, r, r))*r^4+3*(diff(R(r), r))*r-3*(diff(R(r), r, r))*r^2+2*(diff(R(r), r, r, r))*r^3-3*R(r)+4*R(r)*n+2*R(r)*n^2-4*(diff(R(r), r))*r*n+2*(diff(R(r), r))*r*n^2+4*(diff(R(r), r, r))*r^2*n-2*(diff(R(r), r, r))*r^2*n^2-4*R(r)*n^3+R(r)*n^4)/r^4 = (-108-1362*n+2122*n^2+2019*n^3-3032*n^4+401*n^7+1192*n^6-1033*n^5-25*n^9+6*n^10-180*n^8-4128*r^(n+3)*n^3+2304*r^(n+3)*n-576*r^(n+5)*n-760*n^6*r+575*n^5*r-244*n^7*r-2108*n^3*r-11616*r^(n+3)*n^4-5280*r^(n+3)*n^5-456*n^2*r-6912*r^(3*n+3)*n^3+192*r^(n+3)*n^7-6*n^10*r+152*n^8*r+13*n^9*r-192*r^(n+5)*n^7+4224*r^(3*n+3)*n^2-9216*r^(3*n+3)*n^5-16512*r^(3*n+3)*n^4-1632*r^(n+5)*n^6-1536*r^(3*n+3)*n^6-7680*r^(n+5)*n^3-8832*r^(n+5)*n^4-3360*r^(n+5)*n^2-96*r^(n+3)*n^6+2304*r^(3*n+3)*n-5376*r^(n+5)*n^5+2546*n^4*r+4800*r^(n+3)*n^2)/(576*r^4+192*r^4*n^4+1248*r^4*n^3+2688*r^4*n^2+2208*r^4*n):

Sol:=value(dsolve(Eq)):

subs(n=1,Sol);

n:=1: dsolve(Eq);

Conclusion: The decision by value command for any parameters  is incorrect!

## @Markiyan Hirnyk Do you know how ma...

@Markiyan Hirnyk Do you know how many more of these exceptional cases?

Here's another example:

restart:

Eq:=((diff(R(r), r, r, r, r))*r^4+3*(diff(R(r), r))*r-3*(diff(R(r), r, r))*r^2+2*(diff(R(r), r, r, r))*r^3-3*R(r)+4*R(r)*n+2*R(r)*n^2-4*(diff(R(r), r))*r*n+2*(diff(R(r), r))*r*n^2+4*(diff(R(r), r, r))*r^2*n-2*(diff(R(r), r, r))*r^2*n^2-4*R(r)*n^3+R(r)*n^4)/r^4 = (-108-1362*n+2122*n^2+2019*n^3-3032*n^4+401*n^7+1192*n^6-1033*n^5-25*n^9+6*n^10-180*n^8-4128*r^(n+3)*n^3+2304*r^(n+3)*n-576*r^(n+5)*n-760*n^6*r+575*n^5*r-244*n^7*r-2108*n^3*r-11616*r^(n+3)*n^4-5280*r^(n+3)*n^5-456*n^2*r-6912*r^(3*n+3)*n^3+192*r^(n+3)*n^7-6*n^10*r+152*n^8*r+13*n^9*r-192*r^(n+5)*n^7+4224*r^(3*n+3)*n^2-9216*r^(3*n+3)*n^5-16512*r^(3*n+3)*n^4-1632*r^(n+5)*n^6-1536*r^(3*n+3)*n^6-7680*r^(n+5)*n^3-8832*r^(n+5)*n^4-3360*r^(n+5)*n^2-96*r^(n+3)*n^6+2304*r^(3*n+3)*n-5376*r^(n+5)*n^5+2546*n^4*r+4800*r^(n+3)*n^2)/(576*r^4+192*r^4*n^4+1248*r^4*n^3+2688*r^4*n^2+2208*r^4*n):

Sol:=value(dsolve(Eq)):

subs(n=1,Sol);

n:=1: dsolve(Eq);

Conclusion: The decision by value command for any parameters  is incorrect!

Look at this (% - your solution):

subs(n=2, %);

Error, numeric exception: division by zero

Look at this (% - your solution):

subs(n=2, %);

Error, numeric exception: division by zero

## Bug...

Obviously, this is Maple's bug!

## Bug...

Obviously, this is Maple's bug!